Questions — OCR MEI C4 (354 questions)

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OCR MEI C4 2012 January Q4
4 marks Moderate -0.3
  1. Complete the table of values for the curve \(y = \sqrt{\cos x}\).
    \(x\)0\(\frac{\pi}{6}\)\(\frac{\pi}{4}\)\(\frac{3\pi}{8}\)\(\frac{\pi}{2}\)
    \(y\)0.96120.8409
    Hence use the trapezium rule with strip width \(h = \frac{\pi}{8}\) to estimate the value of the integral \(\int_0^{\frac{\pi}{2}} \sqrt{\cos x} \, dx\), giving your answer to 3 decimal places. [3] Fig. 4 shows the curve \(y = \sqrt{\cos x}\) for \(0 \leq x \leq \frac{\pi}{2}\). \includegraphics{figure_4}
  2. State, with a reason, whether the trapezium rule with a strip width of \(\frac{\pi}{16}\) would give a larger or smaller estimate of the integral. [1]
OCR MEI C4 2012 January Q5
5 marks Moderate -0.8
Verify that the vector \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) is perpendicular to the plane through the points A(2, 0, 1), B(1, 2, 2) and C(0, -4, 1). Hence find the cartesian equation of the plane. [5]
OCR MEI C4 2012 January Q6
6 marks Standard +0.3
Given the binomial expansion \((1 + qx)^p = 1 - x + 2x^2 + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]
OCR MEI C4 2012 January Q7
5 marks Moderate -0.3
Show that the straight lines with equations \(\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} -1 \\ 4 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix}\) meet. Find their point of intersection. [5]
OCR MEI C4 2012 January Q8
18 marks Standard +0.3
Fig. 8 shows a cross-section of a car headlight whose inside reflective surface is modelled, in suitable units, by the curve $$x = 2t^2, y = 4t, \quad -\sqrt{2} < t < \sqrt{2}.$$ P\((2t^2, 4t)\) is a point on the curve with parameter \(t\). TS is the tangent to the curve at P, and PR is the line through P parallel to the \(x\)-axis. Q is the point (2, 0). The angles that PS and QP make with the positive \(x\)-direction are \(\theta\) and \(\phi\) respectively. \includegraphics{figure_8}
  1. By considering the gradient of the tangent TS, show that \(\tan \theta = \frac{1}{t}\). [3]
  2. Find the gradient of the line QP in terms of \(t\). Hence show that \(\phi = 2\theta\), and that angle TPQ is equal to \(\theta\). [8]
[The above result shows that if a lamp bulb is placed at Q, then the light from the bulb is reflected to produce a parallel beam of light.] The inside surface of the headlight has the shape produced by rotating the curve about the \(x\)-axis.
  1. Show that the curve has cartesian equation \(y^2 = 8x\). Hence find the volume of revolution of the curve, giving your answer as a multiple of \(\pi\). [7]
OCR MEI C4 2012 January Q9
18 marks Standard +0.3
\includegraphics{figure_9} Fig. 9 shows a hemispherical bowl, of radius 10 cm, filled with water to a depth of \(x\) cm. It can be shown that the volume of water, \(V\) cm\(^3\), is given by $$V = \pi(10x^2 - \frac{1}{3}x^3).$$ Water is poured into a leaking hemispherical bowl of radius 10 cm. Initially, the bowl is empty. After \(t\) seconds, the volume of water is changing at a rate, in cm\(^3\) s\(^{-1}\), given by the equation $$\frac{dV}{dt} = k(20 - x),$$ where \(k\) is a constant.
  1. Find \(\frac{dV}{dx}\), and hence show that \(\pi x \frac{dx}{dt} = k\). [4]
  2. Solve this differential equation, and hence show that the bowl fills completely after \(T\) seconds, where \(T = \frac{50\pi}{k}\). [5]
Once the bowl is full, the supply of water to the bowl is switched off, and water then leaks out at a rate of \(k\) cm\(^3\) s\(^{-1}\).
  1. Show that, \(t\) seconds later, \(\pi(20 - x) \frac{dx}{dt} = -k\). [3]
  2. Solve this differential equation. Hence show that the bowl empties in \(3T\) seconds. [6]
OCR MEI C4 2009 June Q1
7 marks Moderate -0.3
Express \(4\cos\theta - \sin\theta\) in the form \(R\cos(\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Hence solve the equation \(4\cos\theta - \sin\theta = 3\), for \(0 \leq \theta \leq 2\pi\). [7]
OCR MEI C4 2009 June Q2
7 marks Moderate -0.3
Using partial fractions, find \(\int \frac{x}{(x+1)(2x+1)} \, dx\). [7]
OCR MEI C4 2009 June Q3
4 marks Moderate -0.5
A curve satisfies the differential equation \(\frac{dy}{dx} = 3x^2y\), and passes through the point \((1, 1)\). Find \(y\) in terms of \(x\). [4]
OCR MEI C4 2009 June Q4
5 marks Standard +0.3
The part of the curve \(y = 4 - x^2\) that is above the \(x\)-axis is rotated about the \(y\)-axis. This is shown in Fig. 4. Find the volume of revolution produced, giving your answer in terms of \(\pi\). [5] \includegraphics{figure_4}
OCR MEI C4 2009 June Q5
7 marks Standard +0.3
A curve has parametric equations $$x = at^3, \quad y = \frac{a}{1+t^2},$$ where \(a\) is a constant. Show that \(\frac{dy}{dx} = \frac{-2}{3t(1+t^2)^2}\). Hence find the gradient of the curve at the point \((a, \frac{1}{2}a)\). [7]
OCR MEI C4 2009 June Q6
6 marks Standard +0.3
Given that \(\cos\text{ec}^2\theta - \cot\theta = 3\), show that \(\cot^2\theta - \cot\theta - 2 = 0\). Hence solve the equation \(\cos\text{ec}^2\theta - \cot\theta = 3\) for \(0° \leq \theta \leq 180°\). [6]
OCR MEI C4 2009 June Q7
17 marks Standard +0.3
When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point A \((1, 2, 2)\), and enters a glass object at point B \((0, 0, 2)\). The surface of the glass object is a plane with normal vector \(\mathbf{n}\). Fig. 7 shows a cross-section of the glass object in the plane of the light ray and \(\mathbf{n}\). \includegraphics{figure_7}
  1. Find the vector \(\overrightarrow{AB}\) and a vector equation of the line AB. [2]
The surface of the glass object is a plane with equation \(x + z = 2\). AB makes an acute angle \(\theta\) with the normal to this plane.
  1. Write down the normal vector \(\mathbf{n}\), and hence calculate \(\theta\), giving your answer in degrees. [5]
The line BC has vector equation \(\mathbf{r} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ -2 \\ -1 \end{pmatrix}\). This line makes an acute angle \(\phi\) with the normal to the plane.
  1. Show that \(\phi = 45°\). [3]
  2. Snell's Law states that \(\sin\theta = k\sin\phi\), where \(k\) is a constant called the refractive index. Find \(k\). [2]
The light ray leaves the glass object through a plane with equation \(x + z = -1\). Units are centimetres.
  1. Find the point of intersection of the line BC with the plane \(x + z = -1\). Hence find the distance the light ray travels through the glass object. [5]
OCR MEI C4 2009 June Q8
19 marks Standard +0.8
Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).
  1. Fig. 8.1 shows a regular 12-sided polygon inscribed in a circle of radius 1 unit, centre O. AB is one of the sides of the polygon. C is the midpoint of AB. Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \includegraphics{figure_8.1}
    1. Show that AB = \(2\sin 15°\). [2]
    2. Use a double angle formula to express \(\cos 30°\) in terms of \(\sin 15°\). Using the exact value of \(\cos 30°\), show that \(\sin 15° = \frac{1}{4}\sqrt{2 - \sqrt{3}}\). [4]
    3. Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6\sqrt{2 - \sqrt{3}}\). [2]
  2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \includegraphics{figure_8.2}
    1. Show that DE = \(2\tan 15°\). [2]
    2. Let \(t = \tan 15°\). Use a double angle formula to express \(\tan 30°\) in terms of \(t\). Hence show that \(t^2 + 2\sqrt{3}t - 1 = 0\). [3]
    3. Solve this equation, and hence show that \(\pi < 12(2 - \sqrt{3})\). [4]
  3. Use the results in parts (i)(C) and (ii)(C) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form. [2]
OCR MEI C4 2011 June Q1
5 marks Moderate -0.5
Express \(\frac{1}{(2x + 1)(x^2 + 1)}\) in partial fractions. [5]
OCR MEI C4 2011 June Q2
5 marks Moderate -0.8
Find the first three terms in the binomial expansion of \(\sqrt{1 + 3x}\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2011 June Q3
6 marks Moderate -0.3
Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac{1}{2}\pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\). [6]
OCR MEI C4 2011 June Q4
7 marks Moderate -0.3
A curve has parametric equations $$x = 2 \sin \theta, \quad y = \cos 2\theta.$$
  1. Find the exact coordinates and the gradient of the curve at the point with parameter \(\theta = \frac{1}{4}\pi\). [5]
  2. Find \(y\) in terms of \(x\). [2]
OCR MEI C4 2011 June Q5
6 marks Standard +0.3
Solve the equation \(\cosec^2 \theta = 1 + 2 \cot \theta\), for \(-180° \leqslant \theta \leqslant 180°\). [6]
OCR MEI C4 2011 June Q6
7 marks Challenging +1.2
Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at P (1, 2). \includegraphics{figure_6} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7] [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.]
OCR MEI C4 2011 June Q7
18 marks Standard +0.3
A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, G and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \includegraphics{figure_7}
  1. Write down the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). Hence calculate the angle BAC. [6]
  2. Verify that the equation of the plane ABC is \(x + y - 2z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane. [7]
  3. Given that A, B, D and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]
OCR MEI C4 2011 June Q8
18 marks Standard +0.8
Water is leaking from a container. After \(t\) seconds, the depth of water in the container is \(x\) cm, and the volume of water is \(V\) cm\(^3\), where \(V = \frac{1}{3}x^3\). The rate at which water is lost is proportional to \(x\), so that \(\frac{dV}{dt} = -kx\), where \(k\) is a constant.
  1. Show that \(x \frac{dx}{dt} = -k\). [3]
Initially, the depth of water in the container is 10 cm.
  1. Show by integration that \(x = \sqrt{100 - 2kt}\). [4]
  2. Given that the container empties after 50 seconds, find \(k\). [2]
Once the container is empty, water is poured into it at a constant rate of 1 cm\(^3\) per second. The container continues to lose water as before.
  1. Show that, \(t\) seconds after starting to pour the water in, \(\frac{dx}{dt} = \frac{1-x}{x^2}\). [2]
  2. Show that \(\frac{1}{1-x} - x - 1 = \frac{x^2}{1-x}\). Hence solve the differential equation in part (iv) to show that $$t = \ln\left(\frac{1}{1-x}\right) - \frac{1}{2}x^2 - x.$$ [6]
  3. Show that the depth cannot reach 1 cm. [1]
OCR MEI C4 2011 June Q1
1 marks Easy -2.5
In lines 59 and 60, the text says "In that case the proportion suffering such an attack would be 6.4%." Explain how this figure was obtained. [1]
OCR MEI C4 2011 June Q2
5 marks Easy -1.2
  1. In lines 8 to 10, the article says "Some banks do not allow numbers that begin with zero, numbers in which the digits are all the same (such as 5555) or numbers in which the digits are consecutive (such as 2345 or 8765)." How many different 4-digit PINs can be made when all these rules are applied? [3]
  2. At the time of writing, the world population is \(6.7 \times 10^9\) people. Assuming that, on average, each person has one card with a 4-digit PIN (subject to the rules in part (i) of this question), estimate the average number of people holding cards with any given PIN. Give your answer to an appropriate degree of accuracy. [2]
OCR MEI C4 2011 June Q3
2 marks Easy -2.0
In lines 46 and 47, the text says "Of the 11 people with unauthorised transactions, 3 could explain them as breaches of card security (typically losing the card) but 9 could not ... ." Place numbers in the three regions of the diagram consistent with the information in this sentence. [2] \includegraphics{figure_3}